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Theorem supsn 7731
Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
Assertion
Ref Expression
supsn  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )

Proof of Theorem supsn
StepHypRef Expression
1 dfsn2 3902 . . . 4  |-  { B }  =  { B ,  B }
21supeq1i 7709 . . 3  |-  sup ( { B } ,  A ,  R )  =  sup ( { B ,  B } ,  A ,  R )
3 suppr 7730 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  B  e.  A )  ->  sup ( { B ,  B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
433anidm23 1277 . . 3  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B ,  B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
52, 4syl5eq 2487 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
6 ifid 3838 . 2  |-  if ( B R B ,  B ,  B )  =  B
75, 6syl6eq 2491 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ifcif 3803   {csn 3889   {cpr 3891   class class class wbr 4304    Or wor 4652   supcsup 7702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-po 4653  df-so 4654  df-iota 5393  df-riota 6064  df-sup 7703
This theorem is referenced by:  supxrmnf  11292  ramz  14098  xpsdsval  19968  ovolctb  20985  nmoo0  24203  nmop0  25402  nmfn0  25403  esumnul  26514  esum0  26515  ovoliunnfl  28445  voliunnfl  28447  volsupnfl  28448
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